Mass-spring model of elastic media with customizable willis coupling

Willis coupling, in context of acoustics or elasticity designating the coupling between the strain and momentum, have been garnering significant attentions in recent years. As opposed to various applications demonstrated for acoustic wave, elastic media design with Willis coupling on demand is very rare. In this paper, based on a mass-spring model, the accessibility of various components of the coupling tensor for elastic Willis media is explored, and material design with customized Willis coupling aiming to elastic wave control is demonstrated. Homogenization and designability of the model are at first validated via the free wave analysis, then wave transmission properties across a sandwiched Willis layer are analyzed, based on which two illustrative examples for asymmetric reflection and wave mode conversion are demonstrated by specifically designed lattice model. Though the model is conceptual and still far away from practical usage, it may inspire more practical design and further explorations on realizing wave rectification by Willis materials.